## Professor of Mathematics

## Department Chair

Distinguished Research Award, Faculty of Mathematics & Science (2015)

Broadly, my research lies in nonlinear differential equations, symmetry analysis, integrability and solitons, and mathematical physics.

My interests in nonlinear differential equations center on applications of symmetry analysis and conservation laws to the study of PDEs, particularly nonlinear wave equations and soliton equations, as well as ODEs connected with exact solutions through group invariance and other methods. Some applications are also devoted to extending standard methods and developing new approaches both for symmetries and conservation laws. I am co-authoring two books in this area with George Bluman (Department of Mathematics, UBC).

A closely related topic I have been pursuing is finding new conserved integrals and invariants of the equations of fluid flow and magnetohydrodynamics. There are still many interesting open problems in this area.

In mathematical physics, my interests include classical aspects of the Yang-Mills equations, the Einstein gravitational field equations and massless fields on curved spacetime, wave maps (nonlinear sigma/chiral models) and Schrodinger maps (Heisenberg models). One recent focus is on their symmetry and conservation law structure as well as integrable reductions. Another direction for over a decade has been the study of deformations (novel nonlinear generalizations) of gauge theories like Yang-Mills and gravity theories.

A large part of my research in the past 10 years has been geometrical aspects of integrable PDEs. One direction has been classifying integrable hyperbolic systems and group-invariant soliton equations. Related work has studied the geometric origin of such PDEs and their Hamiltonian structure arising from curve flows in various curved generalizations of Euclidean space.

I have also studied soliton interactions for various integrable systems. Recently I have begun working on peakon equations, particularly nonlinear generalizations of the Camassa-Holm equation and the modified Camassa-Holm (FORQ) equation.

**Books and Chapters
**

- G.W. Bluman and S.C. Anco,

Symmetry and Integration Methods for Differential Equations,

Applied Mathematical Sciences, Vol. 154, Springer (New York) 2002.

More information“…for anyone wishing to master techniques for obtaining first integrals of ODEs, this book is outstanding.”

Book Review by Prof. Peter Hydon, UK Nonlinear News, May 2003. - G.W. Bluman, A.F. Cheviakov, S.C. Anco

Applications of Symmetry Methods to Partial Differential Equations,

Applied Mathematical Sciences, Vol. 168, Springer (New York) 2009.

More information“This book is carefully written and provides an excellent overview of this highly active branch of applied mathematics. Like its predecessor, it will be a standard reference in the field for years to come.”

Mathematical Reviews 2561770. - S.C. Anco, Generalization of Noether’s theorem in modern form to non-variational partial differential equations.

In: Recent progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, 119-182,

Fields Institute Communications, Volume 79 (2017).

e-print archive arXiv:math-ph/1605.08734

**Preprints**

- S.C. Anco, E. Asadi,

Hasimoto variables, generalized vortex filament equations, Heisenberg models and Schrodinger maps arising from group-invariant NLS systems

e-print archive arXiv:1901.01879 - S.C. Anco, G. Webb,

Hierarchies of new invariants and conserved integrals in inviscid fluid flow

e-print archive arXiv:1809.01544 - S.C. Anco, E. Recio,

Accelerating dynamical peakons and their behaviour - S.C. Anco, A. Cheviakov

On the different types of global and local conservation laws for partial differential equations in three spatial dimensions

e-print archive arXiv:1803.08859 - S.C. Anco and Z. Yuzbasi+,

Nonlinear integrable systems of Burgers type, Airy type, and Schrodinger type from elastic null curve flows in 3-dimensional Minkowski space

e-print archive arXiv:math-ph/1709.08234 - S.C. Anco and E. Recio+,

A general family of multi-peakon equations and their properties

e-print archive arXiv:math-ph/1609.04354

**Journal Articles
**

**= PhD student; * = MSc student; %= BSc student; += visiting PhD student

- S.C. Anco, M. Gandarias and A. Ballesteros, To appear in Phys. Lett. A.

Global versus local (super)integrability of a nonlinear oscillator

e-print archive arXiv:1809.02248 - S.C. Anco, X. Chang, J. Szmigielski, Stud. Appl. Math. (2018), 1-34

The dynamics of conservative peakons in the NLS hierarchy

e-print archive arXiv:1711.01429 - S.C. Anco, M. Przedborski**, Phys. Rev. E 98 (2018) 042208 (27 pp)

Long wavelength solitary waves in Hertzian chains

e-print archive arXiv:1705.06718 - S.C. Anco and D. Kraus, Discrete and Continuous Dyn. Syst. (Series A) 38(9), (2018), 4449-4465

Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation

e-print archive arXiv:nlin.SI/1708.02520 - S.C. Anco, A. Ahmed*, and E. Asadi, J. Phys. A: Math. Theor. 51 (2018) 065205 (35pp)

Unitarily-invariant integrable systems and geometric curve flows in SU(n+1)/U(n) and SO(2n)/U(n)

e-print archive arXiv:math-ph/1408.5290 - S.C. Anco and E. Recio, J. Phys. A: Math. Theor. 51 (2018) 065203 (19pp)

Conserved norms and related conservation laws for multi-peakon equations

e-print archive arXiv:math-ph/1707.05389 - S.C. Anco and A. Kara, Euro. J. Appl. Math. 29(1) (2018), 78-117

Symmetry invariance of conservation laws

e-print archive arXiv:math-ph/1510.09154 - S.C. Anco and M. Przedborski**, J. Math. Phys. 58 (2017) 091502 (34 pages)

Solitary waves and conservation laws for highly nonlinear wave equations modeling granular chains

e-print archive arXiv:math-ph/1507.04759 - S.C. Anco and F. Mobasheramini*, Physica D 355 (2017), 1-23

Integrable U(1)-invariant peakon equations from the NLS hierarchy

e-print archive arXiv:nlin.SI/1701.00522 - G.M. Webb and S.C. Anco, J. Phys. A: Math. Theor. 50 (2017) 255501 (34pp)

On magnetohydrodynamic gauge field theory

e-print archive arXiv:math-ph/1701.00521 - E. Recio* and S.C. Anco, J. Math. Anal. Appl. 452 (2017) 1229–126

Conservation laws and symmetries of radial generalized nonlinear p-Laplacian evolution equations

e-print archive arXiv:math-ph/1609.07652 - S.C. Anco, Symmetry 9(3) (2017), 33 (28 pages)

On the incompleteness of Ibragimov’s conservation law theorem and its equivalence to a standard formula using symmetries and adjoint-symmetries

e-print archive arXiv:math-ph/1611.02330 - G.M. Webb and S.C. Anco, J. Phys. A: Math. and Theor. 49 (2016) 075501 (48 pages)

Vorticity and symplecticity in multi-symplectic Lagrangian gas dynamics

e-print archive arXiv:math-ph/1601.05031 - S.C. Anco, S. Mohammad*, T. Wolf, C. Zhu+, J. Nonlin. Math. Phys. 23 (2016), 573-606

Generalized negative flows in hierarchies of integrable evolution equations

e-print archive arXiv:nlin.SI/1604.07779 - S.C. Anco, T. Meadows%, V. Pascuzzi%, J. Math. Phys. 57 (2016), 062901 (35 pages)

Some new aspects of first integrals and symmetries for central force dynamics

e-print archive arXiv:math-ph/1508.07258 - S.C. Anco, Int. J. Mod. Phys. B 30 (2016) 1640004 (12 pages)

Symmetry properties of conservation laws

e-print archive arXiv:math-ph/1512.01835 - S.C. Anco and M. Khalique, Int. J. Mod. Phys. B 30 (2016) 1640003 (12 pages)

Conservation laws of coupled semilinear wave equations

e-print archive arXiv:math-ph/1510.09160 - S.C. Anco and K. Alkan*, J. Nonlin. Math. Phys. 23(2) (2016), 256-299

Integrable systems from inelastic curve flows in 2- and 3- dimensional Minkowski space

e-print archive arXiv:nlin.SI/1410.2335 - S.C. Anco, E. Avdonina, A. Gainetdinova+, L. Galiakberova+, N.H. Ibragimov, T. Wolf, J. Phys. A: Math. and Theor. 49 (2016) 105201 (29 pages)

Symmetries and conservation laws of the generalized Krichever-Novikov equation

e-print archive arXiv:nlin.SI/1407.1258 - S.C. Anco, A.S. Mia*, M. Willoughby%, J. Math. Phys. 56 (2015) 121504 (21 pages)

Oscillatory solitons of U(1)-invariant mKdV equations II: Asymptotic behavior and constants of motion

e-print archive arXiv:nlin.SI/1406.6636 - S.C. Anco, A.S. Mia*, M. Willoughby%, J. Math. Phys. 56 (2015) 101506 (35 pages)

Oscillatory solitons of U(1)-invariant mKdV equations I: Envelope speed and temporal frequency

e-print archive arXiv:nlin.SI/1406.6630 - S.C. Anco, A. Dar, N. Tufail+, Proc. Roy. Soc. A (471) 2015, 20150223 (24 pages)

Conserved integrals for inviscid compressible fluid flow in Riemannian manifolds

e-print archive arXiv:math-ph/1503.08859 - S.C. Anco, P. da Silva, I. Freire, J. Math. Phys. 56 (2015) 091506 (21 pages)

A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations

e-print archive arXiv:nlin.SI/1412.4415 - S.C. Anco, N.H. Ibragimov, K.V. Imamutdinova, E.N. Karimova, Applied Math. and Comput. 268 (2015) 52-58

Solutions of gas dynamic equations associated with classical and new conservation laws - S.C. Anco, W. Feng+, T. Wolf, J. Math. Anal. Appl. 427 (2015), 759-786

Exact solutions of semilinear radial Schrodinger equations

by group foliation reduction

e-print archive arXiv:math-ph/1408.3751 - S.C. Anco and W. Feng+, J. Math. Phys. 54 (2013), 121504 (41 pages)

Group-invariant solutions of semilinear Schrodinger equations in multi-dimensions

e-print archive arXiv:math-ph/1301.5529 - S.C. Anco, J. Math. Fluid Mech. 15 (2013), 439-451

New conserved vorticity integrals for moving surfaces in multi-dimensional fluid flow1701.04854

e-print archive arXiv:math-ph/1209.4251 - S.C. Anco and E. Asadi, J. Phys. A: Math. Theor. 45 (2012), 475207 (38 pages)

Symplectically-invariant soliton equations from non-stretching geometric curve flows

e-print archive arXiv:nlin/1206.4040 - S.C. Anco, M. Mohuddin*, and T. Wolf, Appl. Math. Comput. 219 (2012) 679–698

Travelling waves and conservation laws for complex mKdV-type equations

e-print archive arXiv:math-ph/1110.2403 - S.C. Anco, S. MacNaughton*, and T. Wolf, J. Math. Phys. 53 (2012), 053703 (33 pages)

Conservation laws and symmetries of quasilinear radial wave equations in multi-dimensions

e-print archive arXiv:math-ph/1109.1719 - S.C. Anco, N. Tchegoum Ngatat*, and M. Willoughby%, Physica D 240 (2011) 1378-1394

Interaction properties of complex mKdV solitons

e-print archive arXiv:nlin/1102.3620 - S.C. Anco, S. Ali, and T. Wolf, J. Math. Anal. Appl. 379 (2011), 748-763

Symmetry analysis and exact solutions of semilinear heat flow in multi-dimension

e-print archive arXiv:math/1011.4633 - S.C. Anco and R. Myrzakulov, J. Geom. Phys. 60 (2010), 1576-1603

Integrable generalizations of Schrodinger maps and Heisenberg spin models

from Hamiltonian flows of curves and surfaces

e-print archive arXiv:nlin/0806.1360 - S.C. Anco and A. Dar+, Proc. Roy. Soc. A 466 (2010), 2605-2632

Conservation laws of inviscid non-isentropic compressible fluid flow in $n>1$ sp

atial dimensions

e-print archive arXiv:physics.flu-dyn/0911.0882 - S.C. Anco and E. Asadi, J. Phys. A: Math. Theor. 42 (2009), 485201 (25 pages)

Quaternion soliton equations from Hamiltonian curve flows in HP^n

e-print archive arXiv:math-ph/0905.4215 - S.C. Anco and A. Dar+, Proc. Roy. Soc. A 465 (2009), 2461-2488

Classification of conservation laws of compressible isentropic fluid flow in n>1 spatial dimensions

e-print archive arXiv:physics.flu-dyn/0902.3405 - S.C. Anco and S. Vacaru, J. Geom. Phys. 59 (2009), 79-103

Curve flows in Lagrange-Finsler geometry, bi-Hamiltonian structures and solitons

e-print archive arXiv:math-ph/0609070 - S.C. Anco, Inter. J. Theor. Phys. 47 (2008), 684-695

Spinor derivation of quasilocal mean curvature mass in General Relativity

e-print archive arXiv:math-ph/1310.8199 - A. Cheviakov and S.C. Anco, Phys. Lett. A 372 (2008), 1363-1373

Analytical properties and exact solutions of static plasma equilibrium systems in three dimensions

e-print archive arXiv:0708.4247 - S.C. Anco, J. Geom. Phys. 58 (2008), 1-37

Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces

e-print archive arXiv:nlin/0703041 - S.C. Anco, In: IMA Volumes in Mathematics and Its Applications, 144 (2007), 223-250

Hamiltonian curve flows in Lie groups G \subset U(N) and vector NLS, mKdV, sine-Gordon soliton equations

(Proceedings of IMA Workshop on Symmetries and Overdetermined Systems of Partial Differential Equations, 2006)

e-print archive arXiv:nlin/0610075 - S.C. Anco, J. Math. Phys. 48 (2007), 052502 (32 pages)

Mean curvature flow and quasilocal mass for 2-surfaces in Hamiltonian General Relativity

e-print archive arXiv:gr-qc/0402057 - S.C. Anco and N. Ivanova, J. Math. Anal. Appl. 332 (2006), 863-876

Conservation laws and symmetries of semilinear radial wave equations

e-print archive arXiv:math-ph/0608037 - G. Bluman, Temuerchaolu, and S.C. Anco, J. Math. Anal. Appl. 322 (2006), 233-250

New conservation laws obtained directly from symmetry action on known conservation laws

journal archive - S.C. Anco, SIGMA 2 (2006), 044 (18 pages)

Hamiltonian flows of curves in symmetric spaces G/SO(N) and vector soliton equations of mKdV and sine-Gordon type

e-print archive arXiv:nlin/0512046 - S.C. Anco, J. Phys. A: Math. Gen. 39 (2006), 2043-2072

Bi-Hamiltonian operators, integrable flows of curves using moving frames, and geometric map equations

e-print archive arXiv:nlin/0512051 - S.C. Anco and D. The*, Acta Appl. Math. 89 (2005), 1-52

Symmetries, conservation laws, and cohomology of Maxwell’s equations using potentials

e-print archive arXiv:math-ph/0501052 - S.C. Anco and T. Wolf, J. Nonlinear Math. Phys. 12, Supplement 1 (2005), 13-31;

erratum, J. Nonlinear Math. Phys. 12 (2005), 607-608

Some symmetry classifications of hyperbolic vector evolution equations

e-print archive arXiv:nlin/0412015 - S.C. Anco, Inter. J. Geometric Methods in Modern Physics 1 (2004), 493-544

Gauge theory deformations and novel Yang-Mills Chern-Simon field theories with torsion

(Special issue on Advanced Geometric Techniques in Gauge Theory)

e-print archive arXiv:math-ph/0407026 - S.C. Anco and S. Liu, J. Math. Analysis Appl. 297 (2004), 317-342

Exact solutions of semilinear radial wave equations in n dimensions

e-print archive arXiv:math-ph/0309049 - S.C. Anco, Phys. Rev. D 67 (2003), 124007 (17 pages)

Parity violating spin-two gauge theories.

e-print archive arXiv:gr-qc/0305026 - S.C. Anco, J. Phys. A: Math. and Gen. 36 (2003), 8623-8638

Conservation laws of scaling-invariant field equations

e-print archive arXiv:math-ph/0303066 - S.C. Anco, J. Math. Phys. 44 (2003), 1006-1043

Gauge theories of Yang-Mills vector fields coupled to antisymmetric tensor fields

e-print archive arXiv:math-ph/0209051 - S.C. Anco and J. Pohjanpelto, Proc. Roy. Soc. 459 (2003), 1215-1239

Conserved currents of massless spin s fields

e-print archive arXiv:math-ph/0202019 - S.C. Anco, Class. Quant. Grav. 19 (2002), 6445-6467

On multi-graviton and multi-gravitino gauge theories

e-print archive arXiv:gr-qc/0303033 - S.C. Anco, Lett. Math. Phys. 62 (2002), 245-258

Exotic Yang-Mills dilaton gauge theories

e-print archive arXiv:math-ph/0209052 - S.C. Anco and R. Tung, J. Math. Physics 43 (2002), 5531-5566

Covariant Hamiltonian boundary conditions in General Relativity for spatially bounded spacetime regions

e-print archive arXiv:gr-qc/0109013 - S.C. Anco and R. Tung, J. Math. Physics 43 (2002), 3984-4019

Properties of the symplectic structure of General Relativity for spatially bounded spacetime regions

e-print archive arXiv:gr-qc/0109014 - S.C. Anco and G. Bluman, Euro. J. Appl. Math. 13 (2002), 567-585

Direct construction method for conservation laws of partial differential equations II: General treatment

e-print archive arXiv:math-ph/0108024 - S.C. Anco and G. Bluman, Euro. J. Appl. Math. 13 (2002), 545-566

Direct construction method for conservation laws of partial differential equations I: Examples of conservation law classifications

e-print archive arXiv:math-ph/0108023 - S.C. Anco and J. Pohjanpelto, Acta. Appl. Math. 69 (2001), 285-327

Classification of local conservation laws of Maxwell’s equations

e-print archive arXiv:math-ph/0108017 - S.C. Anco and J. Isenberg, Comm. Partial Diff. Eqs. 25 (2000) 1669-1702

Global existence for wave maps with torsion

e-print archive arXiv:math-ph/0007032 - S.C. Anco, Ann. Phys. 270 (1998) 52-125

Nonlinear gauge theories of a spin-two field and a spin-three-halves field

journal archive - S.C. Anco and G. Bluman, Euro. J. Appl. Math. 9 (1998) 245-259

Integrating factors and first integrals of ordinary differential equations

journal archive - S.C. Anco, J. Math. Phys. 38 (1997) 3399-3413

Novel generalization of three dimensional Yang-Mills theory

e-print archive arXiv:math-ph/0209050 - S.C. Anco and G. Bluman, J. Math. Phys. 38 (1997) 3508-3532

Nonlocal symmetries and conservation laws of Maxwell’s equations

journal archive - S.C. Anco and G. Bluman, Phys. Rev. Lett. 78 (1997) 2869-2873

Direct construction of conservation laws from field equations

journal archive - S.C. Anco and G. Bluman, J. Math. Phys. 37 (1996) 2361-2375

Derivation of conservation laws from nonlocal symmetries of differential equations

journal archive - S.C. Anco, J. Math. Phys. 36 (1995) 6553-6565

New spin-one gauge theory in three dimensions

journal archive - S.C. Anco, Phys. Rev. D 50 (1994) 2648-2661

Non-Grassmann generalization of classical supergravity theory

journal archive - S.C. Anco, Contemp. Math. (Amer. Math. Soc.) 132 (1992) 27-50

Construction of locally-symmetric Lagrangian field theories from variational identities

journal archive - S.C. Anco and R.M. Wald, Phys. Rev. D 39 (1989) 2297-2307

Does there exist a sensible quantum theory of an algebra-valued scalar field?

journal archive

**Refereed Conference Proceedings**

- S.C. Anco, M.L. Gandarias, E. Recio, Theor. and Math. Phys. 196(3) (2018), 1241-1259

Conservation laws, symmetries, and line soliton solutions of generalized KP and Boussinesq equations with p-power nonlinearities in two dimensions

(Proceedings of Physics and Mathematics of Nonlinear Phenomena PMNP2017: 50 years of IST, 2017)

e-print archive arXiv:1710.02739 - S.C. Anco, M. Rosa, M.L. Gandarias, Discrete and Continuous Dyn. Syst. (Series S) 11(4) (2018), 607-615

Conservation laws and symmetries of time-dependent generalized KdV equations

(Proceedings of the 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, 2016)

e-print archive arXiv:math-ph/1705.04999 - S.C. Anco, E. Recio+, M.S. Bruzón, AIP Conference Proceedings 1863 (2017), 280002

Conservation laws and potential systems for a generalized thin film equation

(International Conference of Numerical Analysis and Applied Mathematics, 2016) - M.L. Gandarias, M. Rosa, E. Recio+, S.C. Anco, AIP Conference Proceedings 1836 (2017), 020072 (6pp)

Conservation laws and symmetries of a generalized Kawahara equation

(Proceedings of the 1st International Conference on Applied Mathematics and Computer Science, 2017)

e-print archive arXiv:math-ph/1701.04854 - S.C. Anco, E. Recio*, M. Gandarias, M. Bruzon, Dynamical Systems, Differential Equations and Applications (2015), 29-37

A nonlinear generalization of the Camassa-Holm equation with peakon solutions

(Proceedings of the 10th AIMS Conference, 2015)

e-print archive arXiv:math-ph/1609.02473 - S.C. Anco, E. Asadi, A. Dogonchi, Int. J. Mod. Phys. 38 (2015), 1560071 (15pp)

Integrable systems with unitary invariance from non-stretching geometric curve flows in the Hermitian symmetric space Sp(n)/U(n)

(Proceedings of Symmetries, Differential Equations, Applications II, 2014)

e-print archive arXiv:1410.1230 - S.C. Anco, S. Ali, and T. Wolf, SIGMA 7 (2011), 066 (10 pages)

Exact solutions of nonlinear partial differential equations by the method of group foliation reduction

(Proceedings of Conference on Symmetry, Separation, Super-integrability and Special Functions (S4), 2010)

e-print archive arXiv:math-ph/1105.5303 - G. Bluman, A. Cheviakov, and S.C. Anco, In: Group Analysis of Differential Equations and Integrable Systems (2008), 13-35

Construction of conservation laws: How the Direct Method generalizes Noether’s theorem

(Proceedings of 4th International Workshop on Group Analysis of Differential Equations and Integrable Systems, 2007) - S.C. Anco, G. Bluman, and T. Wolf, Acta Appl. Math. 101 (2008), 21-38

Invertible mappings of nonlinear PDEs to linear PDEs through admitted

conservation laws

(Proceedings of International Conference on Geometry and Algebra of PDEs, 2007)

e-print archive arXiv:math-ph/0712.1835 - J. Pohjanpelto and S.C. Anco, SIGMA 4 (2008), 004 (17 pages)

Generalized symmetries of massless free fields on Minkowski space

(Proceedings of the 7th International Conference on Symmetry in Nonlinear Mathematical Physics, 2007)

e-print archive arXiv:math-ph/0801.1892 - S.C. Anco, J.Bland, M. Eastwood, Science in China Series A: Mathematics 49 (2006), 1599-1610

Some Penrose transforms in complex differential geometry

(Proceedings of 6th International Conference on Several Complex Variables, 2006)

journal archive - S.C. Anco and J. Pohjanpelto, In: CRM Proceedings and Lecture Notes, Volume 34 (2004), 1-12

Symmetries and currents of massless neutrino fields, electromagnetic and graviton fields

(Proceedings of Workshop on Symmetry in Physics, 2004)

e-print archive arXiv:math-ph/0306072

**Peakon Equations
**

- Accelerating dynamical peakons and their behaviour (with E. Recio)
- The dynamics of conservative peakons in the NLS hierarchy (with X. Chang, J. Szmigielski)

Studies in Applied Math. (2018), 1-34

e-print archive arXiv: 1711.01429 - Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation (with D. Kraus)

Discrete and Continuous Dynamical Systems (Series A) 38(9), (2018) 4449-4465

e-print archive arXiv:nlin.SI/1708.02520 - A general family of multi-peakon equations and their properties (with E. Recio+)

To appear in J. Phys. A

e-print archive arXiv:math-ph/1609.04354 - Conserved norms and related conservation laws for multi-peakon equations (with E. Recio)

J. Phys. A: Math. Theor. 51 (2018) 065203 (19pp)

e-print archive arXiv:math-ph/1707.05389 - Integrable U(1)-invariant peakon equations from the NLS hierarchy (with F. Mobasheramini*)

Physica D 355 (2017), 1-23

e-print archive arXiv:nlin.SI/1701.00522 - A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations (with P. da Silva, I. Freire)

J. Math. Phys. 56 (2015) 091506 (21 pages)

e-print archive arXiv:nlin.SI/1412.4415 - A nonlinear generalization of the Camassa-Holm equation with peakon solutions (with E. Recio*, M. Gandarias, M. Bruzon)

Dyn. Syst., Diff. Eqns. and Appl. (2015), 29-37

(Proceedings of the 10th AIMS Conference, 2015)

e-print archive arXiv:math-ph/1609.02473

**Conservation Laws and Symmetries of Nonlinear PDEs**

- On the different types of global and local conservation laws for partial differential equations in three spatial dimensions (with A. Cheviakov)

e-print archive arXiv: 1803.08859 - Conserved norms and related conservation laws for multi-peakon equations (with E. Recio)

J. Phys. A: Math. Theor. 51 (2018) 065203 (19pp)

e-print archive arXiv:math-ph/1707.05389 - Symmetry invariance of conservation laws (with A. Kara)

Euro. J. Appl. Math. 29(1) (2018), 78-117

e-print archive arXiv:math-ph/1510.09154 - Conservation laws, symmetries, and line soliton solutions of generalized KP and Boussinesq equations with p-power nonlinearities in two dimensions (with M.L. Gandarias and E. Recio)

Theor. and Math. Phys. 196(3) (2018), 1241-1259

(Proceedings of Physics and Mathematics of Nonlinear Phenomena PMNP2017: 50 years of IST, 2017)

e-print archive arXiv:1710.02739 - Conservation laws and symmetries of time-dependent generalized KdV equations

Discrete and Continuous Dyn. Syst. (Series S) 11(4) (2018), 607-615

(Proceedings of the 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, 2016)

e-print archive arXiv:math-ph/1705.04999 - Conservation laws and symmetries of a generalized Kawahara equation (with M.L. Gandarias, M. Rosa, E. Recio+)

AIP Conference Proceedings 1836 (2017), 020072 (6pp)

(Proceedings of the 1st International Conference on Applied Mathematics and Computer Science, 2017)

e-print archive arXiv:math-ph/1701.04854 - Conservation laws and potential systems for a generalized thin film equation (with E. Recio+, M.S. Bruzón)

AIP Conference Proceedings 1863 (2017), 280002

(International Conference of Numerical Analysis and Applied Mathematics, 2016) - On the incompleteness of Ibragimov’s conservation law theorem and its equivalence to a standard formula using symmetries and adjoint-symmetries

Symmetry 9(3) (2017), 33 (28 pages)

e-print archive arXiv:math-ph/1611.02330 - Conservation laws and symmetries of radial generalized nonlinear p-Laplacian evolution equations (with E. Recio*)

J. Math. Anal. Appl. 452 (2017) 1229–126

e-print archive arXiv:math-ph/1609.07652 - Generalization of Noether’s theorem in modern form to non-variational partial differential equations.

In: Recent progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, 119–182,

Fields Institute Communications, Volume 79 (2017).

e-print archive arXiv:math-ph/1605.08734 - Symmetry properties of conservation laws

Int. J. Mod. Phys. B 30 (2016) 1640004 (12 pages)

e-print archive arXiv:math-ph/1512.01835 - Conservation laws of coupled semilinear wave equations (with M. Khalique)

Int. J. Mod. Phys. B 30 (2016) 1640003 (12 pages)

e-print archive arXiv:math-ph/1510.09160 - Symmetries and conservation laws of the generalized Krichever-Novikov equation (with E. Avdonina, A. Gainetdinova+, L. Galiakberova+, N.H. Ibragimov, T. Wolf)

J. Phys. A: Math. and Theor. 49 (2016) 105201 (29 pages)

e-print archive arXiv:nlin.SI/1407.1258 - Solutions of gas dynamic equations associated with classical and new conservation laws (with N.H. Ibragimov, K.V. Imamutdinova, E.N. Karimova)

Applied Math. and Comput. 268 (2015) 52-58 - Conservation laws and symmetries of quasilinear radial wave equations in multi-dimensions (with S. MacNaughton*,T. Wolf)

J. Math. Phys. 53 (2012), 053703 (33 pages)

e-print archive arXiv:math-ph/1109.1719 - Travelling waves and conservation laws for complex mKdV-type equations (with M. Mohuddin*, T. Wolf)

Appl. Math. Comput. 219 (2012) 679–698

e-print archive arXiv:math-ph/1110.2403 - Construction of conservation laws: How the Direct Method generalizes Noether’s theorem (with G. Bluman, A. Cheviakov)

In: Group Analysis of Differential Equations and Integrable Systems (2008), 13-35

(Proceedings of 4th International Workshop on Group Analysis of Differential Equations and Integrable Systems, 2007) - Invertible mappings of nonlinear PDEs to linear PDEs through admitted

conservation laws (with G. Bluman, and T. Wolf)

Acta Appl. Math. 101 (2008), 21-38

(Proceedings of International Conference on Geometry and Algebra of PDEs, 2007)

e-print archive arXiv:math-ph/0712.1835 - Conservation laws and symmetries of semilinear radial wave equations (with N. Ivanova),

J. Math. Anal. Appl. 332 (2006), 863-876

e-print archive arXiv:math-ph/0608037 - New conservation laws obtained directly from symmetry action on known conservation laws (with G. Bluman, Temuerchaolu)

J. Math. Anal. Appl. 322 (2006), 233-250

journal archive - Symmetries, conservation laws, and cohomology of Maxwell’s equations using potentials (with D. The*)

Acta Appl. Math. 89 (2005), 1-52

e-print archive arXiv:math-ph/0501052 - Symmetries and currents of massless neutrino fields, electromagnetic and graviton fields (with J. Pohjanpelto)

In: CRM Proceedings and Lecture Notes, Volume 34 (2004), 1-12

(Proceedings of Workshop on Symmetry in Physics, 2004)

e-print archive arXiv:math-ph/0306072 - Conserved currents of massless spin s fields (with J. Pohjanpelto)

Proc. Roy. Soc. 459 (2003), 1215-1239

e-print archive arXiv:math-ph/0202019 - Conservation laws of scaling-invariant field equations

J. Phys. A: Math. and Gen. 36 (2003), 8623-8638

e-print archive arXiv:math-ph/0303066 - Direct construction method for conservation laws of partial differential equations II: General treatment (with G. Bluman)

Euro. J. Appl. Math. 13 (2002), 567-585

e-print archive arXiv:math-ph/0108024 - Direct construction method for conservation laws of partial differential equations I: Examples of conservation law classifications (with G. Bluman)

Euro. J. Appl. Math. 13 (2002), 545-566

e-print archive arXiv:math-ph/0108023 - Classification of local conservation laws of Maxwell’s equations (with J. Pohjanpelto)

Acta. Appl. Math. 69 (2001), 285-327

e-print archive arXiv:math-ph/0108017 - Nonlocal symmetries and conservation laws of Maxwell’s equations (with G. Bluman)

J. Math. Phys. 38 (1997) 3508-3532

journal archive - S.C. Anco and G. Bluman, Phys. Rev. Lett. 78 (1997) 2869-2873

Direct construction of conservation laws from field equations

journal archive - Derivation of conservation laws from nonlocal symmetries of differential equations (with G. Bluman)

J. Math. Phys. 37 (1996) 2361-2375

journal archive

**Conserved Integrals and Symmetries in Fluid Mechanics and MHD**

- Hierarchies of new invariants and conserved integrals in inviscid fluid flow (with G. Webb)

eprint archive arXiv:1809.01544 - On magnetohydrodynamic gauge field theory (with G.M. Webb)

J. Phys. A: Math. Theor. 50 (2017) 255501 (34pp)

e-print archive arXiv:math-ph/1701.00521 - Vorticity and symplecticity in multi-symplectic Lagrangian gas dynamics (with G.M. Webb)

J. Phys. A: Math. and Theor. 49 (2016) 075501 (48 pages)

e-print archive arXiv:math-ph/1601.05031 - Conserved integrals for inviscid compressible fluid flow in Riemannian manifolds (with A. Dar, N. Tufail+)

Proc. Roy. Soc. A (471) 2015, 20150223 (24 pages)

e-print archive arXiv:math-ph/1503.08859 - New conserved vorticity integrals for moving surfaces in multi-dimensional fluid flow

J. Math. Fluid Mech. 15 (2013), 439-451

e-print archive arXiv:math-ph/1209.4251 - Conservation laws of inviscid non-isentropic compressible fluid flow in n>1 spatial dimensions (with A. Dar+)

Proc. Roy. Soc. A 466 (2010), 2605-2632

e-print archive arXiv:physics.flu-dyn/0911.0882 - Classification of conservation laws of compressible isentropic fluid flow in n>1 spatial dimensions (with A. Dar+)

Proc. Roy. Soc. A 465 (2009), 2461-2488

e-print archive arXiv:physics.flu-dyn/0902.3405 - Analytical properties and exact solutions of static plasma equilibrium systems in three dimensions (with A. Cheviakov)

Phys. Lett. A 372 (2008), 1363-1373

e-print archive arXiv:0708.4247

**Exact Solutions and Symmetry Analysis of Nonlinear PDEs and ODEs**

- Global versus local (super)integrability of a nonlinear oscillator (with M. Gandarias and A. Ballesteros)

To appear in Phys. Lett. A

e-print archive arXiv:1809.02248 - Conservation laws, symmetries, and line soliton solutions of generalized KP and Boussinesq equations with p-power nonlinearities in two dimensions (with M.L. Gandarias and E. Recio)

Theor. and Math. Phys. 196(3) (2018), 1241-1259

(Proceedings of Physics and Mathematics of Nonlinear Phenomena PMNP2017: 50 years of IST, 2017)

e-print archive arXiv:1710.02739 - Conservation laws and symmetries of radial generalized nonlinear p-Laplacian evolution equations (with E. Recio*)

J. Math. Anal. Appl. 452 (2017) 1229–126

e-print archive arXiv:math-ph/1609.07652 - Exact solutions of semilinear radial Schrodinger equations

by group foliation reduction (with W. Feng+, T. Wolf)

J. Math. Anal. Appl. 427 (2015), 759-786

e-print archive arXiv:math-ph/1408.3751 - Group-invariant solutions of semilinear Schrodinger equations in multi-dimensions (with W. Feng+)

J. Math. Phys. 54 (2013), 121504 (41 pages)

e-print archive arXiv:math-ph/1301.5529 - Travelling waves and conservation laws for complex mKdV-type equations (with M. Mohuddin*, T. Wolf)

Appl. Math. Comput. 219 (2012) 679–698

e-print archive arXiv:math-ph/1110.2403 - Symmetry analysis and exact solutions of semilinear heat flow in multi-dimension (with S. Ali, T. Wolf)

J. Math. Anal. Appl. 379 (2011), 748-763

e-print archive arXiv:math/1011.4633 - Exact solutions of nonlinear partial differential equations by the method of group foliation reduction (with S. Ali, T. Wolf)

SIGMA 7 (2011), 066 (10 pages)

(Proceedings of Conference on Symmetry, Separation, Super-integrability and Special Functions (S4), 2010)

e-print archive arXiv:math-ph/1105.5303 - Generalized symmetries of massless free fields on Minkowski space (with J. Pohjanpelto)

SIGMA 4 (2008), 004 (17 pages)

(Proceedings of the 7th International Conference on Symmetry in Nonlinear Mathematical Physics, 2007)

e-print archive arXiv:math-ph/0801.1892 - Exact solutions of semilinear radial wave equations in n dimensions (with S. Liu)

J. Math. Analysis Appl. 297 (2004), 317-342

e-print archive arXiv:math-ph/0309049 - Integrating factors and first integrals of ordinary differential equations (with G. Bluman)

Euro. J. Appl. Math. 9 (1998) 245-259

journal archive

**Solitons, Solitary waves, and Interactions**

- Long wavelength solitary waves in Hertzian chains (M. Przedborski**)

Phys. Rev. E 98 (2018) 042208 (27 pp)

e-print archive arXiv:1705.06718 - Solitary waves and conservation laws for highly nonlinear wave equations modeling granular chains (with M. Przedborski**)

J. Math. Phys. 58 (2017) 091502 (34 pages)

e-print archive arXiv:math-ph/1507.04759 - Oscillatory solitons of U(1)-invariant mKdV equations II: Asymptotic behavior and constants of motion (with A.S. Mia*, M. Willoughby%)

J. Math. Phys. 56 (2015) 121504 (21 pages)

e-print archive arXiv:nlin.SI/1406.6636 - Oscillatory solitons of U(1)-invariant mKdV equations I: Envelope speed and temporal frequency (with A.S. Mia*, M. Willoughby%)

J. Math. Phys. 56 (2015) 101506 (35 pages)

e-print archive arXiv:nlin.SI/1406.6630 - Travelling waves and conservation laws for complex mKdV-type equations (with M. Mohuddin*, T. Wolf)

Appl. Math. Comput. 219 (2012) 679–698

e-print archive arXiv:math-ph/1110.2403 - Interaction properties of complex mKdV solitons (with N. Tchegoum Ngatat*, M. Willoughby%)

Physica D 240 (2011) 1378-1394

e-print archive arXiv:nlin/1102.3620

**Integrable Systems and Geometric Curve Flows**

- Hasimoto variables, generalized vortex filament equations, Heisenberg models and Schrodinger maps arising from group-invariant NLS systems (witrh E. Asadi)

e-print archive arXiv:1901.01879 - Elastic null curve flows, nonlinear C-integrable systems, and geometric realization of Cole-Hopf transformations (with Z. Yuzbasi+)

e-print archive arXiv:math-ph/1709.08234 - Unitarily-invariant integrable systems and geometric curve flows in SU(n+1)/U(n) and SO(2n)/U(n) (with A. Ahmed*, E. Asadi)

J. Phys. A: Math. Theor. 51 (2018) 065205 (35pp)

e-print archive arXiv:math-ph/1408.5290 - Generalized negative flows in hierarchies of integrable evolution equations (with S. Mohammad*, T. Wolf, C. Zhu+)

J. Nonlin. Math. Phys. 23 (2016), 573-606

e-print archive arXiv:nlin.SI/1604.07779 - Integrable systems from inelastic curve flows in 2- and 3- dimensional Minkowski space (with K. Alkan*)

J. Nonlin. Math. Phys. 23(2) (2016), 256-299

e-print archive arXiv:nlin.SI/1410.2335 - Integrable generalizations of Schrodinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces (with R. Myrzakulov)

J. Geom. Phys. 60 (2010), 1576-1603

e-print archive arXiv:nlin/0806.1360 - Symplectically-invariant soliton equations from non-stretching geometric curve flows (with E. Asadi)

J. Phys. A: Math. Theor. 45 (2012), 475207 (38 pages)

e-print archive arXiv:nlin/1206.4040 - Quaternion soliton equations from Hamiltonian curve flows in HP^n (with E. Asadi)

J. Phys. A: Math. Theor. 42 (2009), 485201 (25 pages)

e-print archive arXiv:math-ph/0905.4215 - Curve flows in Lagrange-Finsler geometry, bi-Hamiltonian structures and solitons (with S. Vacaru)

J. Geom. Phys. 59 (2009), 79-103

e-print archive arXiv:math-ph/0609070 - Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces

J. Geom. Phys. 58 (2008), 1-37

e-print archive arXiv:nlin/0703041 - Hamiltonian curve flows in Lie groups G \subset U(N) and vector NLS, mKdV, sine-Gordon soliton equations

In: IMA Volumes in Mathematics and Its Applications, 144 (2007), 223-250

(Proceedings of IMA Workshop on Symmetries and Overdetermined Systems of Partial Differential Equations, 2006)

e-print archive arXiv:nlin/0610075 - Hamiltonian flows of curves in symmetric spaces G/SO(N) and vector soliton equations of mKdV and sine-Gordon type

SIGMA 2 (2006), 044 (18 pages)

e-print archive arXiv:nlin/0512046 - Bi-Hamiltonian operators, integrable flows of curves using moving frames, and geometric map equations

J. Phys. A: Math. Gen. 39 (2006), 2043-2072

e-print archive arXiv:nlin/0512051 - Some symmetry classifications of hyperbolic vector evolution equations (with T. Wolf)

J. Nonlinear Math. Phys. 12, Supplement 1 (2005), 13-31; erratum, J. Nonlinear Math. Phys. 12 (2005), 607-608

e-print archive arXiv:nlin/0412015

**Deformations of Classical Gauge Field Theories**

- Gauge theory deformations and novel Yang-Mills Chern-Simon field theories with torsion

Inter. J. Geometric Methods in Modern Physics 1 (2004), 493-544

(Special issue on Advanced Geometric Techniques in Gauge Theory)

e-print archive arXiv:math-ph/0407026 - Parity violating spin-two gauge theories

Phys. Rev. D 67 (2003), 124007 (17 pages)

e-print archive arXiv:gr-qc/0305026 - Gauge theories of Yang-Mills vector fields coupled to antisymmetric tensor fields

J. Math. Phys. 44 (2003), 1006-1043

e-print archive arXiv:math-ph/0209051 - On multi-graviton and multi-gravitino gauge theories

Class. Quant. Grav. 19 (2002), 6445-6467

e-print archive arXiv:gr-qc/0303033 - Exotic Yang-Mills dilaton gauge theories

Lett. Math. Phys. 62 (2002), 245-258

e-print archive arXiv:math-ph/0209052 - Nonlinear gauge theories of a spin-two field and a spin-three-halves field

Ann. Phys. 270 (1998) 52-125

journal archive - Novel generalization of three dimensional Yang-Mills theory

J. Math. Phys. 38 (1997) 3399-3413

e-print archive arXiv:math-ph/0209050 - New spin-one gauge theory in three dimensions

J. Math. Phys. 36 (1995) 6553-6565

journal archive - Non-Grassmann generalization of classical supergravity theory

Phys. Rev. D 50 (1994) 2648-2661

journal archive - Construction of locally-symmetric Lagrangian field theories from variational identities

Contemp. Math. (Amer. Math. Soc.) 132 (1992) 27-50

e-print archive arXiv: 1711.01429

**Mathematical Physics and General Relativity**

- Global versus local (super)integrability of a nonlinear oscillator (with M. Gandarias and A. Ballesteros)

To appear in Phys. Lett. Ae-print archive arXiv:1809.02248 - Some new aspects of first integrals and symmetries for central force dynamics (with T. Meadows^%, V. Pascuzzi^%)

J. Math. Phys. 57 (2016), 062901 (35 pages)

e-print archive arXiv:math-ph/1508.07258 - Generalized symmetries of massless free fields on Minkowski space (with J. Pohjanpelto)

SIGMA 4 (2008), 004 (17 pages)

(Proceedings of the 7th International Conference on Symmetry in Nonlinear Mathematical Physics, 2007)

e-print archive arXiv:math-ph/0801.1892 - Spinor derivation of quasilocal mean curvature mass in General Relativity

Inter. J. Theor. Phys. 47 (2008), 684-695

e-print archive arXiv:math-ph/1310.8199 - Mean curvature flow and quasilocal mass for 2-surfaces in Hamiltonian General Relativity

J. Math. Phys. 48 (2007), 052502 (32 pages)

e-print archive arXiv:gr-qc/0402057 - Some Penrose transforms in complex differential geometry (with J.Bland, M. Eastwood)

Science in China Series A: Mathematics 49 (2006), 1599-1610

(Proceedings of 6th International Conference on Several Complex Variables, 2006)

journal archive - Covariant Hamiltonian boundary conditions in General Relativity for spatially bounded spacetime regions (with R. Tung)

J. Math. Physics 43 (2002), 5531-5566

e-print archive arXiv:gr-qc/0109013 - Properties of the symplectic structure of General Relativity for spatially bounded spacetime regions (with R. Tung)

J. Math. Physics 43 (2002), 3984-4019

e-print archive arXiv:gr-qc/0109014 - Global existence for wave maps with torsion (with J. Isenberg)

Comm. Partial Diff. Eqs. 25 (2000) 1669-1702

e-print archive arXiv:math-ph/0007032 - Does there exist a sensible quantum theory of an algebra-valued scalar field? (with R.M. Wald)

Phys. Rev. D 39 (1989) 2297-2307

journal archive

- MATH 3P09 Introduction to Partial Differential Equations
- MATH 3P51 Applied Mathematics with MAPLE
- MATH/PHYS 4P09 and 5P09 Solitons and Nonlinear Wave Equations
- MATH/PHYS 4P94 and PHYS 5P65 General Relativity and Black Holes
- MATH 5P60 Partial Differential Equations

**PhD**

Dmitry Pshenitsin (Theoretical Physics), June 2016

**MSc**

Richard (SIcheng) Zhao (Mathematics), in progress

Jordan Fazio (Theoretical Physics), in progress

James Frank (Theoretical Physics), in progress

Evans Boadi (Mathematics), 2018

Elise Kinnear (Mathematics), 2017

Sarah Bax (Mathematics), 2017

Fataneh Mobasheramini (Mathematics), 2016

Elena Recio (Mathematics), 2016

Shahid Mohammad (Mathematics), 2015

Ahmed Ahmed (Mathematics), 2014

Kivilcim Alkan (Mathematics), 2012

Sattar Mia (Mathematics), 2012

Nestor Tchegoum Ngatat (Mathematics), 2011

Steve MacNaughton (Mathematics), 2011

Scott Greenhalgh (Mathematics), 2008

**Research**

Michelle Przedborski (PhD, Theoretical Physics, Brock), 2017

Elena Recio (PhD, Mathematics, University of Cadiz), 2017

Aliya Gainetdinova (PhD, Mathematics, Ufa State Technical University), 2015

Liaisan Galiakberova (PhD, Mathematics, Ufa State Technical University), 2015

Zuhal Kucukarslan Yuzbasi (PhD, Mathematics, Firat University), 2014

Nazim Tufail (PhD, Mathematics, Quaid-e-Azam University), 2014

Wei Feng (PhD, Mathematics, Northwest University (China), 2013

Amanullah Dar (PhD, Mathematics, Quaid-e-Azam University), 2010

Elena Karimova, (MSc, Mathematics, Ufa State Technical University), 2015

Kseniya Imamutdinova (MSc, Mathematics, Ufa State Technical University), 2014

Jordan Fazio (USRA), 2016

Sarah Bax (USRA), 2014

Michel Grenier (USRA), 2011

Andrae Blanchard (USRA), 2007

Andrew Copfer (USRA), 2002